ST(i) = MAT + D [ AT(i) - MAT ]
Where: ST(i) is the mean Stream Temperature for day i,
MAT is the Mean Air Temperature for that year,
D is a dampening coefficient, and
AT(i) is the mean Air Temperature of day i.
The daily air temperature and its annual mean are obtained from the air temperature model. The dampening coefficient is a station specific factor that can be estimated directly from the data and possibly also from various other data layers. The dampening coefficient is unconstrained, but it is expected to range from near 0 for springs which emerge near the mean air temperature yearlong, to near 1 for streams that mimic the daily air temperature very closely. Using the July 1 through September 15 data from the 96 time series available, The best fit D ranges from 0.2037 to 0.9655 with the median of 0.4560. For the 6 stations that have 2 years of data, the mean absolute difference in the paired D's is only 0.0436, even while the 12 raw D's range from 0.2037 to 0.7129. This offers some confirmation that D is a station specific parameter.
Initial evaluation of the stream temperature model indicated that stream temperature was somewhat too responsive to daily air temperature and that a lag factor was suggested. The model was therefore modified so that the stream temperature for day i is obtained as before, but using a weighted average of the air temperature from today (i), yesterday (i-1), and the day before yesterday (i-2). The weights used are the best fit to the entire data using an exponential decay. The first 2 days of the year are calculated without lag terms, but remember that the current model is intended for July 1 through September 15 only. With additional data, this lag and smoothing function could become another model parameter, but for now, it is treated as a constant. So we now have:
ST(i) = MAT + 0.62413 D [ AT(i) - MAT ]
+ 0.26411 D [ AT(i-1) - MAT ]
+ 0.11176 D [ AT(i-2) - MAT ]
Finally, the predicted stream temperature is limited to a minimum of 0.0°C (32.0°F). Due to ice formation, stream temperatures seldom fall much below freezing.